Credibility For Sale∗
Harris Dellas†
Dirk Niepelt‡
March 4, 2014
Abstract
We develop a sovereign debt model with official and private creditors where the
probability of default depends on both the level and the composition of liabilities.
Higher exposure to official lenders improves incentives to repay but carries extra
costs, due to overcharging or reduced ex-post flexibility. The model implies that
official lending to sovereigns takes place in times of debt distress; carries a favorable
rate; and can displace private funding even under pari passu provisions. Moreover,
in the presence of long-term debt overhang, the availability of official funds increases
the probability of default on existing debt although default does not trigger exclusion
from private credit markets. These findings help shed light on joint default and debt
composition choices such as those observed during the recent sovereign debt crisis
in Europe.
JEL class: F34, H63
Keywords: Sovereign debt, official lending, default, enforcement
1
Introduction
The recent sovereign debt crisis in the Eurozone exhibited a rich set of patterns: A
heavily indebted country, Greece, defaulted on its debt while at the same time receiving
substantial official funds (from European sources as well as the IMF) carrying a low
interest rate. Greece’s default decision was encouraged by its official creditors. Other
countries, namely Ireland, Portugal and Spain, did not default but the composition of
their new loans (including commitments by the Eurogroup to Spain) shifted significantly
∗
For useful comments and conversations, we thank Marios Angeletos, Tobias Broer, Fernando Broner,
Fabrice Collard, Behzad Diba, Martı́n Gonzalez-Eiras, Leo Martinez, Andreas Schabert, Jean Tirole as
well as conference and seminar participants at the Federal Reserve Bank of Chicago/NBER Conference
“Macroeconomics Within and Across Borders”, EIEF Rome and ESSIM 2013.
greece73.tex
†
Department of Economics, University of Bern, CEPR. VWI, Schanzeneckstrasse 1, CH-3012 Bern,
Switzerland. Phone: +41 (0)31-631-3989. [email protected], www.harrisdellas.net.
‡
Study Center Gerzensee, University of Bern, IIES, Stockholm University, CEPR. P.O. Box 21, CH3115 Gerzensee, Switzerland. [email protected], alum.mit.edu/www/niepelt.
1
towards official sources. That is, in spite of pari passu provisions on new private loans,
official funds crowded out private sources of financing. Yet other countries including Italy
opted to abstain from official funding all together and relied on more expensive private
loans.
To account for these phenomena, we propose a simple extension of the standard
sovereign debt model (Eaton and Gersovitz, 1981) that addresses default decisions, the
choice of composition of sovereign debt between private and official sources and the interaction between these two decisions. We introduce two plausible and realistic features
(motivated below): A class of official creditors who possess superior enforcement power
because they can impose more severe sanctions in the case of default than private creditors;1 and a pari passu provision that prevents ex-post discrimination against private
lenders. With these extensions, the standard model provides important insights on the
phenomena described above.
A sovereign borrower that lacks commitment only repays her debt when the cost of
default is sufficiently large. This limits the amount the country can borrow. In order
to procure more funds ex ante, the sovereign may want to structure its debt in a way
that increases default sanctions ex post. Exposure to official lenders accomplishes this
and under pari passu, it strengthens the sovereign’s credibility both vis-a-vis official and
private creditors. But borrowing from official lenders also carries costs. On the one hand,
official funds may be more expensive than private funds on a risk-adjusted basis.2 On the
other hand, official lending may limit ex-post flexibility. For more severe default sanctions
imply that debt will be repaid in some states of the world where the sovereign would have
defaulted if the debt had been issued to private creditors instead.3
The model has the following implications: First, countries typically find it optimal to
borrow from private sources when financing needs are low. When these needs are high,
sovereigns favor stronger reliance on official sources of funding because the benefits of
credibility outweigh the costs associated with those funds. The resulting variation in the
debt composition does not arise because of variation in risk-adjusted interest rates—these
are constant over the debt cycle. It rather reflects the fact that the sovereign’s cost of
borrowing includes additional components, namely expected social losses from default and
lack of consumption smoothing, and that the magnitude of these components varies with
total borrowing needs.4
1
That the identity—official vs. private—of the creditor can make a difference for sovereign risk assessments appears rather undisputed. IMF lending is a case in point. But the identity of the creditor also
seems to matter in private loan transactions. According to a widely shared presumption and anecdotal
evidence the incentive to repay loans to Mafia is much stronger than the incentive to repay other creditors,
due to Mafia’s more extensive set of enforcement tools.
2
For example, official rates may contain a surcharge over private lending rates to cover administrative
and other costs, as is the case with IMF loans. Or the rate on official loans may contain a “penalty”
component as was the case during the Eurozone debt crisis.
3
See Zame (1993) for a discussion of the insurance benefits of implicitly state contingent debt. Under
incomplete markets, a country may trade a higher interest rate on its debt for the option to declare
default in states where debt repayment would have been very costly.
4
In a benchmark case without mark-down on official funds (no rent extraction by official lenders)
official and private loans have identical risk and return characteristics. Nevertheless, the government is
not indifferent to the debt composition because this composition affects the risk and return characteristics
2
Second, large official loans may go hand in hand with interest rates that look inexpensive in the sense that private debt of the same amount would result in much higher interest
rates. This is a direct consequence of strengthened credibility. Whether the interest rate
reduction is large depends on factors that determine the bargaining positions of official
lenders and the sovereign.
Third, when the hedging benefits of private—and thus riskier—loans are negligible,
debt quotas tend to be higher under official funding. However, default risk tends to be
equal or lower when funds are borrowed from official sources, as intended, unless the
credibility gains are small and creditors are willing to subsidize the loans.
Fourth, while pari passu provisions succeed in strengthening credibility vis-a-vis both
official and private lenders they may fail to induce private investors to continue providing
significant, fresh funds. That is, official financing may still crowd out private funding.
Crowding out is less likely when the credibility gains of official intervention or the price
elasticity of debt are large.
Finally, the preferred debt composition depends on the current default decision and,
if the sovereign opts against default, on the stock of outstanding long-term debt (among
other factors). The default decision, in turn, depends on the stock of debt overhang and
on the sources of refinancing available to the borrowing country. With long-term debt
overhang, the availability of official credit makes current default more likely in spite of the
fact that default does not trigger exclusion from private credit markets. Official lenders
may not only tolerate default on private long-term debt in this case but actually encourage
it because they can earn rents by refinancing the country.5 If high costs discourage current
default, however, larger debt overhang makes it more likely that a country shies away from
official funds.
These model implications square well with patterns observed during the recent debt
crisis in the Eurozone. The case of Greece (with a high debt overhang) conforms with the
prediction that in debt distress periods, default is more likely when the debt overhang
is large and that when it occurs, it is accompanied by a switch from private towards
official sources of funding. The cases of Italy, Ireland, Portugal and Spain (in decreasing
order of debt overhang) are broadly consistent with the model implication that absent
a default, the share of official funds in fresh borrowing depends negatively on the stock
of outstanding long-term debt.6 The fact that relatively low interest rates have been
negotiated for official loans suggests strong bargaining power on the part of the borrowing
countries.7
of both assets.
5
That is, the sovereign and his official would be creditors may gang up against existing, private holders
of sovereign debt.
6
IMF (2012, p. 17) contains debt quota data. While Spain has not yet received official funding the
Eurogroup committed to provide up to 100 billion Euro. There exist other explanations for why a debt
distressed country shuns official funds, for example that the provision of such funds may be accompanied
by demands for politically costly reforms. Our analysis offers a complementary explanation that does not
rely on political economy considerations.
7
According to observers, the IMF is repositioning itself in favor of a more “aggressive” stance visa-vis holders of outstanding private debt, “seeking earlier and deeper haircuts of government creditors”
(Financial Times, June 5, 2013, Creditors Likely Losers From IMF Sovereign Rethink). If true, such
3
The first key new element in our model relative to the standard sovereign debt setup
concerns the existence of differential enforcement powers across groups of creditors. Such
differential powers may arise for various reasons. One reason is that the credit relationship
is part of a broader arrangement between the borrower and the lender, as it is typically the
case from participation in the same club. Consider for instance the relationship between
Greece and the other members of the Eurozone. A Greek default on official loans from
those countries could trigger retaliation and lower Greece’s benefits from club membership
in the European Monetary Union (EMU) or even the European Union (EU): Structural
fund payments and other transfers might be cut; Greece might be forced to leave the
Eurozone; official lenders might be tempted to adopt policies that are less favorable to
Greek interests; support for Greek foreign policy positions might wither; and so on.
As the ongoing crisis constitutes the first instance in which certain members of the
Eurozone have borrowed large amounts from other members, and since no default against
official funds has occurred we cannot yet know whether official lenders would be in a
position to inflict sanctions of the type described above and if they were, whether they
would actually choose to do so.8 But what matters for the behavior of agents in our
model—and hence for the properties of equilibrium—is the perception of the existence
and likely use of such sanctioning powers, rather than the use itself.9 In our view, the
public debate in Europe and statements by policy makers provide ample evidence for a
widely shared belief that superior sanctioning powers do exist and official lenders would
be willing to use them.
For example, in Germany which provides most of the official financing, the statements
of politicians, the debates in parliament and the public reaction all conjure the impression
that the loans are perceived to face a low probability of default. In fact, such a perception
was a sina qua non for large German loan provision at low rates to be politically feasible in
the first place, given voters’ expressed antipathy to solidarity (transfers) towards Greece.
This perception is also founded in the knowledge that a default by Greece on debt held
by official creditors amounts to violating EU treaties and breaking national laws, leaving
Greece in uncharted and treacherous political territory regarding its future within the
EU.10 Naturally, time consistency is an issue as it would also be costly for Germany
to impose sanctions ex post. But repeat business within the club (lending to Portugal,
Ireland and Spain is but one example) makes reputational considerations important, and
not imposing sanctions following a Greek default could undermine Germany’s credibility.11
repositioning would seem to be consistent with the model.
8
Superior power certainly existed during the times when mighty countries would use military force to
enforce repayment (for instance, when the British navy bombarded Athens).
9
Naturally, in a model with asynchronous borrowing and default decisions of multiple borrowers,
default by one country could reveal the existence of such powers and affect perceptions in those countries
that have not made a default decision yet.
10
The German government spokesman Steffen Seibert argued that the countries of the Eurozone could
not accept a reduction in the value of their loans to Greece because this would contradict European Union
treaties as well as national legislation in Germany and other countries that prohibits member countries
to assume the debts of other countries (Kathimerini, November 27, 2012).
11
Steffen Seibert has argued that debt forgiveness would lead to a huge loss of credibility for Germany
and could encourage other countries with debt problems to ask for similar treatment (Kathimerini,
4
Note also that in order to ensure broad political support for enforcement ex post, Germany
has required club-wide participation in the official lending operations.
Similar perceptions about the additional, severe cost of Greek default as a consequence
of Eurozone loans are also held in Greece: Voters have opted for parties that strongly
oppose default and warn about its dire consequences for Greece’s membership in EMU
and even EU. While the main opposition party advocates default on both private and
official loans this position may not reflect the view that official lenders are powerless but
rather an underlying desire to actually subject Greece to the enforcement and get the
country expelled from EMU.12 In our view, the evidence thus points to a widely shared
belief in both the existence of superior sanctioning powers on the part of official lenders
and their willingness to use them.
The second key new element in the model, the pari passu provision, also fits the
European experience. Anxious to avoid a crowding out of private funding, official lenders
conceded that safeguards should be put in place to impede ex-post discrimination against
their private counterparts. Consistent with this intention, the Greek debt exchange in
Spring 2012 put private and official lenders (the EFSF) on an equal footing,13 and more
recent discussions about financial support for Spain revealed similar considerations.14 In
another context and looking forward, the New York court case between Argentina and
Elliott Management might end up undermining the preferred creditor status of certain
lenders and further contribute to sovereign debt arrangements characterized by pari passu.
Related Literature and Outline The literature on the composition of sovereign debt
by type of creditor is scant. Boz (2011) reviews the literature on IMF lending, summarizes
empirical evidence and presents a quantitative model of a sovereign that may borrow from
private lenders and the IMF. In her model private lending is subject to default risk while
IMF lending is default risk free. The cost of IMF funds exceeds the risk free rate by an
exogenous surcharge. It predicts modest, countercyclical and intermittent IMF lending
with complete crowding out of private funding. We think that the last feature makes
the model unsuitable for analyzing the recent Euro debt crisis as private and official new
lending co-exist and this is a key fact that needs to be explained.15
November 27, 2012).
12
The main opposition party is called the “party of the drachma” in the sense that special interest
groups that support it—such as heavily indebted press barons, labor unions in the public sector, professional guilds etc.—are thought to profit from Greece’s exit from European institutions.
13
Zettelmeyer, Trebesch and Gulati (2012, p. 25) report that “Greece and the remaining signatories of
the agreement committed to a payment schedule in which the EFSF and bondholders would be repaid
pro-rata and on the same day. In the event of a shortfall in payments by Greece, the common paying
agent committed to distributing allocating this shortfall pro rata between the EFSF and the bondholders.
Hence, the co-financing agreement makes it difficult for Greece to default on its bondholders without also
defaulting on the EFSF.”
14
The Wall Street Journal (June 29, 2012, Investors Cheer Europe Deal) reports that Angela Merkel’s
agreement “to make ESM loans to Spain equal to Spanish bonds in creditors’ pecking order was largely
a recognition by Germany that this was necessary to protect Spain’s ability to sell bonds . . . .”
15
Boz’s model also has the unsatisfactory property that accessing the IMF lending facilities changes
preferences in the sense that it triggers an increase in the sovereign’s discount factor.
5
Bolton and Jeanne (2011) analyze the interaction between multiple sovereigns of different credit quality and the banking system in a financially integrated area. They argue
that a country issuing ‘safe haven’ government debt may derive rents from exploiting its
position as monopolistic supplier of this safe asset. In the model proposed here, we also
allow for non-competitive rents, but in contrast to Bolton and Jeanne (2011), we consider the possibility that (official) lenders rather than the borrower extract rents. Niepelt
(2011) analyzes the composition of sovereign debt across maturities rather than lenders,
as considered here, and Diamond and He (2012) analyze the implications of the maturity
structure of debt overhang on investment decisions. Finally, Tirole (2012) distinguishes
between ex-post bailouts that aim at avoiding collateral damage and ex-ante risk-sharing
(for example joint-and-several liability) among sovereigns.
The rest of the paper is organized as follows. The model is set up in section 2. The
equilibrium is characterized in section 3. Section 4 contains a series of tractable examples
that help develop intuition and illustrate the main results. Section 5 concludes.
2
Model
The economy lasts for two periods, t = 1, 2. It is inhabited by a representative taxpayer,
a government and foreign investors. Taxpayers neither save nor borrow.16 They have
time- and state-additive preferences over consumption with strictly increasing and concave
felicity function u(·) and discount factor δ ∈ (0, 1). Welfare of taxpayers in period t = 1
is given by
#
" 2
X
E
δ t−1 u(ytp − τt )|s1 , π1 ,
t=1
ytp
where
denotes exogenous, pre-tax income, τt taxes, st the state (to be specified below)
and πt the policy choice in period t. We often write E1 [·] instead of E[·|s1 , π1 ].
Foreign investors are risk neutral, require a risk free gross interest rate β −1 > 1 and
hold all government debt (since taxpayers do not save).17 To guarantee positive debt
positions, we assume δ β as is standard in the sovereign debt literature.18 Foreign
investors are composed of private and official lenders. Private lenders are competitive.
Official lenders—we refer to them as “the enforcer”—may coordinate amongst themselves
and behave non-competitively vis-a-vis the borrowing country. Either as a consequence of
this, or due to differences in the cost of funds across lenders, the interest rate charged by
official lenders may differ from that charged by private lenders. The central implications
of the model are independent of this feature.
The government maximizes the welfare of taxpayers. In period t, it chooses the repayment rate on maturing debt, rt , issues zero-coupon, one-period debt, bt+1 , of which bet+1 is
16
Mankiw (2000) or Matsen, Sveen and Torvik (2005) analyze fiscal policy in economies with “savers”
and “spenders.”
17
The assumption that the sets of taxpayers and investors do not “overlap” simplifies the analysis and
does not matter for the main results.
18
For recent examples, see Aguiar and Gopinath (2006) or Arellano (2008).
6
held by the enforcer and bt+1 −bet+1 by private lenders, and (residually) levies taxes. Without loss of generality, public spending other than debt repayment is normalized to zero.
Crucially, the government cannot commit its successors (or future selves). Short-sales are
ruled out.
Let b02 denote the stock of debt issued to private investors in the past that is due in
period 2. Define b̃2 ≡ b02 ξ1 + b2 to be the stock of maturing debt in period t = 2, where ξ1
is a variable linked to the default decision in the first period: If default in the first period
applies to debt maturing in that period and also to the outstanding long-term debt, then
ξ1 ≡ r1 . If, instead, default in the first period does not affect the repayment rate on
long-term debt, then ξ1 ≡ 1 and b̃2 = b02 + b2 . While the latter specification is consistent
with a strict notion of lack of commitment, the former often seems plausible on the basis
of legal and economic grounds and it also generates more closely intertwined default and
refinancing choices. We solve the model under either specification. Outstanding long-term
debt may be repurchased by the government in period t = 1. The short-sale constraints in
the first period therefore read b2 −be2 ≥ −b02 ξ1 and be2 ≥ 0 or, more compactly, b̃2 ≥ be2 ≥ 0.
Let B(b02 ξ1 ) denote the set of debt ownership structures (b2 , be2 ) that are consistent with
the two short-sale constraints.
A sovereign default—a situation where the repayment rate falls short of unity—triggers
income losses for taxpayers (see Eaton and Gersovitz, 1981; Cole and Kehoe, 2000; Aguiar
and Gopinath, 2006; Arellano, 2008). More specifically, a default in period t triggers
an income loss Lt ≥ 0 where Lt is the realization of an i.i.d. random variable with
cumulative distribution function Ft (·) and associated density function ft (·), ft (L) > 0 for
all Lt ≥ 0.19 In the presence of official lending, default triggers additional income losses
for the borrowing country (see the discussion in the introduction). These losses are given
by L(be2 ) with L(0) = 0 and L0 (be2 ) ≥ 0 for all be2 > 0.20 Default occurs uniformly across
privately and officially held debt (see the discussion on pari passu in the introduction).
The sequence of events in each period is as follows. In the beginning of period t, Lt
and yt become known. The state is given by st = (yt , Lt , b̃t , bet ). Conditional on st , the
government chooses policies, π1 = (r1 , b2 , be2 ) or π2 = r2 , taking as given the equilibrium
relationship between these choices and bond prices.
Let q1 (s1 , π1 ) and p1 (s1 , π1 ) denote the period t = 1 state s1 price of debt issued to
private and official lenders, respectively, if the government implements policy π1 . When
choosing its policy, the government takes the price functions q1 (s1 , ·) and p1 (s1 , ·) as given.
Letting ∆1 (s1 , π1 ) ≡ q1 (s1 , π1 ) − p1 (s1 , π1 ) denote the difference between the two prices
we define the borrowing country’s deficit in period t = 1 as
d1 (s1 , π1 ) ≡ b2 q1 (s1 , π1 ) − be2 ∆1 (s1 , π1 ).
(1)
The budget constraint of the government is τ1 = b1 r1 − d1 (s1 , π1 ). The pre-tax income
of taxpayers is y1p = y1 − 1[r1 <1] L1 and y2p = y2 − 1[r2 <1] (L2 + L(be2 )) where 1[x] denotes
the indicator function for event x. Taxpayers’ consumption therefore is given by c1 =
19
Default costs are sometimes assumed to depend on income, Lt (yt ). Our specification allows for
arbitrary correlation between yt and Lt .
20
Our results would effectively remain unchanged under different assumptions about the function L(·).
For example, L(·) could depend on be2 /b2 rather than be2 .
7
y1 − b1 r1 − 1[r1 <1] L1 + d1 (s1 , π1 ) in the first period and c2 = y2 − b̃2 r2 − 1[r2 <1] (L2 + L(be2 ))
in the second period.
Let G1 (s1 ) denote the value of the government’s program conditional on state s1 and
let G1 (s1 ; π1 ) denote the value conditional on a particular first-period policy choice. We
have
G1 (s1 ) =
G2 (s2 ) =
max
r1 ∈[0,1], (b2 ,be2 )∈B(b02 ξ1 )
u(y1 − b1 r1 − 1[r1 <1] L1 + d1 (s1 , π1 )) + δE1 [G2 (s2 )]
s.t. p1 (s1 , ·), q1 (s1 , ·),
max u(y2 − b̃2 r2 − 1[r2 <1] (L2 + L(be2 ))).
r2 ∈[0,1]
The government chooses the repayment rate on maturing debt as well as debt issuance in
period t = 1 in order to maximize the sum of the flow utility from consumption in that
period as well as the discounted expected continuation value. The latter represents the
maximized flow utility from consumption in period t = 2, as reflected by the second value
function. Importantly, the default rate in period t = 2 is chosen by the government in
that period alone, due to the lack of commitment.
An equilibrium conditional on the official-funds price function p1 (st , ·) then consists of
value and policy functions in periods t = 1 and t = 2 and a private-funds price function
q1 (st , ·) such that
i. conditional on s1 as well as the price functions, the policy choices are optimal for
the borrowing country,
πt (st ) solves Gt (st ), t = 1, 2;
ii. the private-funds price function reflects rational expectations as well as the participation constraint of competitive private lenders (i.e., investors earn the expected,
competitive rate of return),
q1 (s1 , π1 ) = β E1 [r2 (s2 )] .
(2)
Note that equilibrium is defined conditional on a price function for official funds,
p1 (·, ·). This allows us to study debt policy under alternative assumptions about the
institutional environment in place and the enforcer’s cost of funds. Consider for example
the case in which the enforcer has negligible bargaining power. In this case, the equilibrium
price p1 (s1 , π1 ) is set so that the enforcer attains no more than his outside option. If
exposure to the borrowing country after a default generates some costs C(be2 ) (beyond
capital losses) to the enforcer then the enforcer’s binding participation constraint implies
be2 p1 (s1 , π1 ) = βbe2 E1 [r2 (s2 )] − β Prob[r2 (s2 ) < 1] C(be2 ).
(3)
As another example, consider the case where the enforcer has sufficient bargaining
power vis-a-vis the borrowing country to negotiate a fixed “mark-down” relative to the
price on private markets. The equilibrium price of official funds then equals
p1 (s1 , π1 ) = κ q1 (s1 , π1 ), 0 < κ < 1.
8
(4)
In both examples, p1 (s1 , π1 ) ≤ q1 (s1 , π1 ).21
We proceed under the assumption that the government’s program is well behaved and
gives rise to smooth policy functions. In the examples considered below, we verify that
this is indeed the case.
3
Analysis
Choice of Repayment Rate in the Second Period Consider first the government’s
choice of repayment rate in the last period, r2 . Since the marginal cost of lowering r2 is
zero when r2 < 1, the optimal repayment rate equals either zero or unity. In particular,
1 if L2 ≥ b̃2 − L(be2 )
r2 (s2 ) =
.
(5)
0 if L2 < b̃2 − L(be2 )
Condition (5) states that the government chooses to default when the resulting income
losses, L2 + L(be2 ), are smaller than the amount of debt coming due.22 Condition (5) is
consistent with the notion that governments tend to default when the political costs—
specifically income losses of pivotal pressure groups—are low. Governments also tend
to default when economic activity is depressed (Borensztein, Levy Yeyati and Panizza,
2006; Tomz and Wright, 2007). The model is consistent with this fact as well when it
is slightly extended to include direct default costs for the government in addition to the
income losses for taxpayers. Note that corner solutions for the optimal repayment rate
follow under more general assumptions about default costs than those made here.
Equation (5) pins down the expected repayment rate. From (2), the equilibrium price
of private funds equals
q1 (s1 , π1 ) = β(1 − F2 (b̃2 − L(be2 )))
(6)
and is decreasing in the quantity of debt issued, b2 . If b02 > 0 and ξ1 = r1 then the choice
of repayment rate in the first period, r1 , also affects the price because it determines b̃2 .
We return to this point later, when discussing the equilibrium choice of r1 .
Choice of Debt Issued to Private Lenders Issuing debt to private lenders has two
effects on the deficit. On the one hand, it raises funds from the marginal unit of debt, in
21
It is also possible to think of situations where κ > 1, for example because official lenders subsidize
borrowing in order to account for externalities.
22
Letting bp2 denote privately held debt and r2p , r2e the repayment rates on bp2 and be2 , respectively, our
assumption in the main model corresponds to the case where
c2 = y2 − be2 r2e − bp2 r2p − 1[r2e ·r2p <1] L2 − 1[r2e ·r2p <1] L(be2 ).
An alternative specification,
c2 = y2 − be2 r2e − bp2 r2p − 1[r2e ·r2p <1] L2 − 1[r2e <1] L(be2 ),
could give rise to selective default against private lenders (as long as be2 < L(be2 )) but never to selective
default against official lenders. Boz (2011) completely rules out default against the enforcer.
9
proportion to its price. On the other hand, it reduces the funds raised from inframarginal
units of private and official lending, by changing the price of these units. This latter
effect is a direct consequence of the government’s lack of commitment and reflects the
endogeneity of subsequent repayment decisions. Formally, from (1) and (6),
∂d1 (s1 , π1 )
∂q1 (s1 , π1 )
∂∆1 (s1 , π1 )
= q1 (s1 , π1 ) + b2
− be2
∂b2
∂b2
∂b2
∂∆1 (s1 , π1 )
= q1 (s1 , π1 ) − b2 βf2 (b̃2 − L(be2 )) − be2
.
∂b2
Funding from private sources is maximized at the top of the “debt-Laffer curve” which
is reached when the above marginal effect equals zero. A completely myopic government
(δ = 0) maximizes the deficit and attains the maximum of the debt-Laffer curve. A
non-myopic government (δ > 0), in contrast, does not maximize the deficit because each
additional unit of debt strictly reduces the continuation value. In either case, the equilibrium value of b2 is therefore (weakly) smaller than the value that attains the maximum of
the debt-Laffer curve. Moreover, this equilibrium value (weakly) exceeds be2 − b02 ξ1 , due
to the short-sale constraint vis-a-vis private investors (b2 − be2 ≥ −b02 ξ1 ). In the following,
we refer to the range of b2 values defined by the lower bound of be2 − b02 ξ1 and the upper
bound of the maximizer of the debt-Laffer curve as the “relevant range” for b2 .
Let λ and µ denote the multipliers associated with the short-sale constraints be2 ≥ 0
and b2 ≥ be2 −b02 ξ1 , respectively. The effect of a marginal increase in debt issued to private
lenders on the government’s objective is given by
∂d1 (s1 , π1 )
∂E1 [G2 (s2 )]
∂G1 (s1 ; π1 )
= u0 (c1 )
+δ
+µ
∂b2
∂b2
∂b2
which can be expressed as23
(1 − F2 (b̃2 − L(be2 )))(βu0 (c1 ) − δE1 [u0 (y2 − b̃2 )])
0
e
e ∂∆1 (s1 , π1 )
+ µ.
− u (c1 ) b2 βf2 (b̃2 − L(b2 )) + b2
∂b2
(7)
The first part of this marginal effect represents the consumption smoothing benefit
from the marginal unit of debt. It differs from the corresponding expression in the case
without default risk because the price of debt equals β(1 − F2 (b̃2 − L(be2 ))) rather than
β and because debt repayment occurs with probability (1 − F2 (b̃2 − L(be2 ))) rather than
23
We use the fact that
∂E1 [G2 (s2 )]
=
∂b2
Z b̃2 −L(be2 )
Z ∞
∂
∂
E1 [u(y2 − L2 − L(be2 ))|L2 ]dF2 (L2 ) +
E1 [u(y2 − b̃2 )]dF2 (L2 )
∂b2 0
∂b2 b̃2 −L(be2 )
= E1 [u(y2 − b̃2 )]f2 (b̃2 − L(be2 )) − E1 [u(y2 − b̃2 )]f2 (b̃2 − L(be2 )) − (1 − F2 (b̃2 − L(be2 )))E1 [u0 (y2 − b̃2 )].
10
always.24 The marginal rate of substitution between current and future consumption
and thus, the profile of output, as well as the relative price between current and future
consumption determine the strength of the consumption smoothing benefit.
The second part of the marginal effect arises because the repayment probability depends on the quantity issued: Each extra unit of debt issued lowers the price of all
inframarginal units or, equivalently, raises the interest rate on them. This increase in the
interest rate—which would be absent in a model with commitment—makes first period
consumption more expensive. As a consequence, the equilibrium amount of debt issued
(conditional on be2 ) tends to be smaller than that under commitment. The second part also
reflects the fact that issuance of b2 might affect the price difference ∆1 (s1 , π1 ). The final
part of the marginal effect, the multiplier µ, is strictly positive if the short-sale constraint
b2 ≥ be2 − b02 ξ1 is binding, and equals zero otherwise.
It may seem surprising that the negative welfare effect associated with the reduction
of funds raised from inframarginal units of debt (the second part discussed above) is not
balanced by a positive welfare effect from the reduced repayment probability of these
inframarginal units in the future. In fact, this effect is present. However, it does not
appear in (7) because it is equal in absolute value to a third welfare effect of opposite
sign, reflecting the increased risk of future social losses in the wake of default.25 It is these
social losses that are at the source of the reduced incentive (relative to the commitment
case) for the government to issue debt. Niepelt (2011) contains a detailed discussion in
the context of a model with multiple maturities.
Choice of Debt Issued to Official Lenders Issuing debt to official lenders while
holding total debt constant (that is, substituting official for private debt) affects the
deficit threefold. First, by raising the output losses of the borrowing country in case of
future default, it reduces default risk and increases the price of debt. This has a positive
effect on the deficit. Second, it reduces the deficit at the margin by the amount ∆1 (s1 , π1 )
if private creditors purchase debt at a higher price than official lenders. Finally, it may
change the price discount applied on the inframarginal units of debt issued to the enforcer.
Formally, from (1) and (6),
∂∆1 (s1 , π1 )
∂d1 (s1 , π1 )
= b2 βf2 (b̃2 − L(be2 ))L0 (be2 ) − ∆1 (s1 , π1 ) − be2
.
e
∂b2
∂be2
The effect of substituting official for private funds on the government’s objective is
given by
∂G1 (s1 ; π1 )
∂d1 (s1 , π1 )
∂E1 [G2 (s2 )]
= u0 (c1 )
+δ
+λ−µ
e
e
∂b2
∂b2
∂be2
With risk free debt, the marginal effect would reduce to βu0 (c1 ) − δE1 [u0 (y2 − b̃2 )].
Higher debt issuance increases subsequent default risk and thus, the risk of future output losses in
the wake of default. The corresponding first-order welfare effects that operate through the continuation
value are zero. This is a consequence of an envelope condition—the subsequent government is indifferent
at the margin between bearing the costs of debt repayment on the one hand or income losses in the wake
of default on the other (see footnote 23).
24
25
11
where the multipliers reflect the two short-sale constraints. This can be expressed as26
"Z
#!
e
b̃2 −L(b2 )
L0 (be2 ) u0 (c1 )βf2 (b̃2 − L(be2 ))b2 − δE1
u0 (y2 − L2 − L(be2 ))dF2 (L2 )
0
0
e ∂∆1 (s1 , π1 )
−u (c1 ) ∆1 (s1 , π1 ) + b2
+ λ − µ.
∂be2
(8)
The first part of this marginal effect reflects the benefit of stronger credibility on the
one hand and the cost of reduced flexibility on the other. A larger share of official debt
generates stronger repayment incentives and hence lower default risk; this raises q1 (s1 , π1 )
and the deficit, and it allows the country to consume more in the first period. But the
larger share of official debt also inflicts additional income losses in case default nevertheless
occurs subsequently which happens for low realizations of L2 . These income losses hurt
particularly if marginal utility is high in default states. Conditional on the distribution
F2 , the cost of reduced flexibility therefore is a more important concern when income y2
negatively correlates with L2 such that default states are states with low consumption.
The second part of the marginal effect reflects the price difference between private and
official loans, as well as the fact that changing the debt composition may affect this price
difference.
Choice of Repayment Rate in the First Period The trade-off governing the choice
of r1 differs depending on whether ξ1 = 1 or ξ1 = r1 . Consider first the latter case. When
ξ1 = r1 (and b02 > 0), then the trade-off governing the choice or r1 is a dynamic one
because default does not only wipe out maturing debt but also long-term debt overhang.
This reduces the probability of default in the second period and raises the price q1 (s1 , π1 ).
For r1 < 1, the net marginal benefit of reducing the repayment rate further is positive
and as a consequence, the optimal repayment rate equals either zero or unity. Letting
G1 (s1 ; r1 = ρ) denote the value of the government’s program conditional on state s1 and
repayment rate r1 = ρ, the equilibrium choice thus satisfies
1 if G1 (s1 ; r1 = 1) ≥ G1 (s1 ; r1 = 0)
r1 (s1 ) =
.
(9)
0 if G1 (s1 ; r1 = 1) < G1 (s1 ; r1 = 0)
We discuss the interdependence between long-term debt overhang and the default decision
(9) in more detail below.
If ξ1 = 1, in contrast, default wipes out maturing debt, b1 , but not outstanding longterm debt, b02 . The choice of r1 therefore does not affect the price of debt paid by private
lenders, q1 (s1 , π1 ). If the same holds true for the price paid by official lenders, p1 (s1 , π1 ),
(and thus, the price difference ∆1 (s1 , π1 )) then the deficit is independent of r1 as well and
the repayment decision in the first period parallels the one in the second period, namely
1 if L1 ≥ b1
r1 (s1 ) =
.
0 if L1 < b1
26
Note that ∂E1 [G2 (s2 )]/∂be2 = −L0 (be2 )E1
hR
b̃2 −L(be2 )
0
12
i
u0 (y2 − L2 − L(be2 ))dF2 (L2 ) (see footnote 23).
Independence of p1 (s1 , ·) and r1 if ξ1 = 1 may be a reasonable assumption in some
environments but not in others. The assumption is satisfied in the particular specifications
discussed above (see equations (3) and (4)) where the trade-offs present in the lending
relationship between the enforcer and the borrowing country from period t = 1 onwards
are independent of the repayment rate r1 . But it would not be satisfied if the enforcer’s
participation constraint held “before” r1 were chosen. Equation (3) then would be replaced
by
be2 p1 (s1 , π1 ) = βbe2 E1 [r2 (s2 )] − β Prob[r2 (s2 ) < 1] C(be2 ) + be1 r1
and the equilibrium price of debt purchased by the enforcer would depend on the repayment rate in the first period. In this case, the enforcer would be indifferent between
lowering r1 by an amount and increasing p1 by the amount be1 /be2 . Such a combination of
changes in r1 and p1 could strictly increase the welfare of the borrowing country if be1 < b1 ,
that is, if there were another group of investors that could be “burned.”27 Consequently,
a default in the first period could be in the joint interest of the borrowing country and
the enforcer.28 We do not pursue this variation of the model here.
Properties of Equilibrium The equilibrium conditions make clear that the quantity
of debt issued, the ownership structure, and the default choices depend on factors such
as the intensity of the borrowing needs, as manifested in the ratio β/δ and the steepness
of the output profile; the distribution function of output losses, F2 (·); preferences; the
enforcement technology, L(·); and the price discount, ∆1 (·). Since the general model
cannot be solved in closed form, the exact contribution of these factors is difficult to
identify. Nevertheless, some general conclusions can be drawn, in particular relating to
the effects of pari passu on the equilibrium debt ownership structure and to the interaction
between this structure and the default decision in the first period.
Conditions (7) and (8) imply that four types of equilibria may emerge. Letting
M(b2 , be2 ) and Me (b2 , be2 ) denote the marginal effects without multipliers in (7) and (8),
respectively, these four types can be summarized as follows:
i. µ = λ = 0. b2 , be2 interior with M(b2 , be2 ) = Me (b2 , be2 ) = 0.
ii. µ = 0, λ > 0. b2 interior, be2 = 0 with M(b2 , 0) = 0, Me (b2 , 0) < 0.
iii. µ > 0, λ = 0. b2 = be2 > 0 with M(b2 , b2 ) + Me (b2 , b2 ) = 0.
iv. µ > 0, λ > 0. b2 = be2 = 0 with M(b2 , b2 ) + Me (b2 , b2 ) < 0.
Suppose for simplicity that the mark-down on official funds is constant, 1−∆1 (s1 , π1 )/q1 (s1 , π1 ) =
κ, and normalize (7) and (8) by the direct contribution of a marginal unit of debt to utility
27
Naturally, a proper specification of the problem would require that such incentives are recognized
and priced ex ante.
28
Broner, Martin and Ventura (2010) argue that secondary markets undermine the ability of a sovereign
to discriminate between groups of lenders. The above argument suggests that the borrowing country may
collude with lenders rolling over its debt and discriminate against other holders of outstanding debt by
choosing r1 and p1 appropriately.
13
in the first period, u0 (c1 )β(1 − F2 (b̃2 − L(be2 ))). This yields
δE1 [u0 (cnd
2 )]
e
(10)
1−
− (b2 − be2 (1 − κ))H2 (b̃2 − L(be2 )),
M(b2 , b2 ) ∝
βu0 (c1 )
Me (b2 , be2 ) ∝ L0 (be2 ) ×
(11)
i
hR

e
b̃ −L(b2 ) 0 d
δE1 0 2
u (c2 )dF2 (L2 )
(b2 − be2 (1 − κ))H2 (b̃2 − L(be2 )) −

βu0 (c1 )(1 − F2 (b̃2 − L(be2 )))
−(1 − κ).
Here, H2 (·) denotes the hazard function, H2 (·) ≡ f2 (·)/(1 − F2 (·)), and cd2 and cnd
2 denotes consumption in default and non-default states, respectively. In (10), the term in
parentheses represents the consumption smoothing benefit from the marginal unit of debt
and the second term the negative price effect due to higher debt issuance, weighted by
the quantity of debt. In (11), the term in parentheses represents the marginal benefit of
credibility net of the cost of reduced flexibility, and the last term reflects the reduction in
the deficit due to the marginal substitution towards marked-down official funds.
Under standard assumptions, the hazard function is weakly increasing. As a consequence, the debt-Laffer curve is hump shaped and for any given value of be2 , a unique value
for b2 solves the condition M(b2 , be2 ) = 0.29 Moreover, this value for b2 tends to rise as be2
increases.30 This is a direct consequence of pari passu in the sense that higher be2 improves
the price elasticity of official and private funds.31 But the model makes clear that this
feature may not prevent crowding out of private funds—in contrast to what appears to be
intended (see the discussion in the introduction). In fact, such crowding out takes place
whenever b2 increases by less than one-to-one with be2 . This is more likely to be the case
when the functions L(·) and H2 (·) are relatively flat.32
Pari passu also crucially affects the cost benefit analysis of official funds as represented
by Me (b2 , be2 ). For the pari passu restriction to put official and private lenders on an equal
footing at the time of repayment implies that strengthened credibility raises the prices of
both official and private funds (as reflected in the first term of (11)). This gives rise to
a complementarity between total and official lending that parallels the complementarity
in (10) although it works in the opposite direction: While the complementarity in (11)
29
The hazard function is closely related to the price elasticity of debt, for b2 /q1 ∂q1 /∂b2 = −b2 H2 .
Examples of distribution functions with increasing hazard functions include uniform, normal, exponential,
logistic, extreme value, Laplace, power, Weibull, gamma, chi-squared, chi, or beta distributions (see, e.g.,
Bagnoli and Bergstrom, 2005). Abstracting from official lending, the deficit is given by β(1 − F2 )b2 and
its derivative with respect to b2 equals β(1 − F2 )(1 − b2 H2 ). With a weakly increasing hazard function,
this derivative strictly decreases as b2 increases (at least up to the top of the debt-Laffer curve).
30
Suppose for simplicity that marginal utility is constant such that an interior b2 is pinned down by
the condition 1 − δ/β = (b2 − be2 (1 − κ))H2 (b02 + b2 − L(be2 )). If L(·) and H2 (·) are increasing and κ ≤ 1
then a rise of be2 leads b2 to increase as well.
31
Without pari passu, a higher quantity of official loans is likely to increase default risk on private
loans, depending on the exact relationship between the then, two default choices and default costs.
32
For example, if F (·) is exponential such that the hazard function is constant and if marginal utility
is constant as well then an increase of be2 leaves the equilibrium value of b2 unaltered. Official lending
then fully crowds out private lending.
14
arises because a higher quantity of total borrowing increases the benefit of strengthened
credibility, the complementarity in (10) occurs because a higher quantity of official lending
affects the debt-Laffer curve in a favorable way. Higher b2 therefore increases the marginal
benefit of be2 while higher be2 increases the net benefit of total borrowing, and it is precisely
the pari passu restriction that is at the source of these complementarities.
Absent sufficient curvature of u(·), F (·) or L(·) the complementarities may give rise
to corner solutions for the debt composition. This is confirmed by the analysis of the
special cases of the model discussed below. Numerical simulations of the model under
general functional form assumptions suggest that although interior debt compositions are
not unusual, the forces that push the debt composition into a corner in the fully linear
case seem to operate strongly also in the general case.
Turning to the interaction between the composition of fresh funds and the default
decision in the first period, consider the role of long-term debt overhang, b02 ξ1 > 0. For
given values of b2 and be2 , long-term debt overhang has two consequences for the marginal
effects M(b2 , be2 ) and Me (b2 , be2 ). On the one hand, it lowers the price of newly issued
debt (and thus, the deficit) and changes the elasticity of the price, as is evident from the
fact that the density functions F2 (·) and f2 (·) in (10) and (11) depend on b̃2 . This affects
the marginal benefit of both b2 and be2 . On the other hand, long-term debt overhang
increases the marginal expected cost of enforcer funds, due to reduced flexibility in the
R b̃ −L(be2 ) 0 d
future, as reflected by the term −L0 (be2 ) 0 2
u (c2 )dF2 (L2 ) in (11). Long-term debt
overhang therefore may reduce the attractiveness of official relative to private funding, and
precisely for this reason, it may encourage default when refinancing from official sources
is particularly desirable. We discuss this in more detail in one of the following examples.
4
Examples
In order to characterize equilibrium in closed form and present solutions that shed light
on the first-order determinants of default choices and the debt ownership structure, we
abstract from all non-essential sources of non-linearity.33 (At the end of the section, we
briefly discuss the consequences of relaxing the linearity assumption.) In particular, we
let u0 (c) = 1, L0 (be2 ) = L0 with 0 ≤ L0 < 1, and F20 (L2 ) = f2 over the relevant range.34
This implies (net of some constants)
G1 (s1 ) =
max
(Z
r1 ∈[0,1], (b2 ,be2 )∈B(b02 ξ1 )
b̃2 −L0 be2
−δ
−b1 r1 − 1[r1 <1] L1 + β(1 − f2 · (b̃2 − L0 be2 ))b2 − ∆1 (s1 , π1 )be2
)
(L2 + L0 be2 )f2 dL2 + (1 − f2 · (b̃2 − L0 be2 ))b̃2
.
(12)
0
We highlight the role played by the intensity of borrowing needs, β/δ, enforcement
power, L0 , the price discount, ∆(·), and long-term debt overhang, b02 ξ1 , by working
33
This comes at the cost of ignoring the hedging benefits of privately held (and thus, riskier) debt.
The restriction L0 < 1 is required for a debt-Laffer curve to exist. Without it, official lending could
completely eliminate default risk. The analysis and implications of the case of L0 ≥ 1 are straightforward.
Note that if L0 ≥ 1 and ∆1 (s1 , π1 ) = 0, then the country attains the commitment outcome.
34
15
through a series of examples. These examples illustrate that the model can account
for the issuance of debt to official lenders in periods of debt distress (and no such issuance
in other periods) at yields that appear favorable to the borrower compared with the yields
that would have to be paid on private markets. They also illustrate the general properties
of equilibrium discussed earlier, including the interaction between debt overhang, default
and the composition of new borrowing.
Exogenous Price Discount, No Long-Term Debt Overhang Suppose that funds
provided by the enforcer carry an exogenous, constant price discount relative to funds
obtained from private investors, p1 (s1 , π1 ) = κq1 (s1 , π1 ) with κ ≤ 1, and let b02 = 0. The
constant price discount implies ∆1 (s1 , π1 ) ≡ q1 (s1 , π1 )(1 − κ) and the marginal effects
defined earlier equal
M(b2 , be2 ) = (1 − F2 )(β − δ) − βf2 (b2 − be2 (1 − κ)),
Me (b2 , be2 ) = L0 (βf2 (b2 − be2 (1 − κ)) − δF2 ) − β(1 − F2 )(1 − κ)
(13)
(14)
where F2 ≡ f2 · (b2 − L0 be2 ) denotes the probability of default. Holding be2 constant, G1
is concave in b2 . The determinant of the Hessian is negative, and the Hessian thus is
indefinite.35 This implies that any interior critical point of (12) constitutes a saddle point
such that the equilibrium necessarily is in a corner. We consider the two interesting
corner equilibria—one with private debt and the other with official debt—in turn. The
uninteresting third corner equilibrium with zero debt is ruled out by assuming that δ/β
is sufficiently small.
If sovereign debt is exclusively funded from private sources then M(b2 , 0) = 0. Solving
for the equilibrium yields the following values for the debt levels and the government’s
program in the first period (for δ ≤ β):36
bPR
2 =
1 (β − δ)2
1 β−δ
, be2 PR = 0, GPR
.
=
1
f2 2β − δ
2f2 2β − δ
The maximum of the debt-Laffer curve is obtained at the debt level (2f2 )−1 , the level
chosen by a myopic government with δ = 0, and is associated with a default probability
of 1/2.
If instead all debt is funded from official sources then M(b2 , b2 ) + Me (b2 , b2 ) = 0 and
the equilibrium values (for δ ≤ βκ) are given by
bOF
2 =
βκ − δ
1
1
(βκ − δ)2
1
1
e OF
OF
OF
,
b
=
b
,
G
=
.
2
2
1
0
0
0
f2 2βκ − δ(1 − L ) 1 − L
2f2 2βκ − δ(1 − L ) 1 − L0
The maximum of the debt-Laffer curve is now obtained at the debt level (2f2 (1 − L0 ))−1
and is again associated with a default probability of 1/2. For L0 > 0, the debt level
attaining the maximum of the debt-Laffer curve is higher in the corner with official than
with private debt.
In the special case of 1 − κ = L0 the determinant is zero. See, for example, Simon and Blume (1994,
Theorem 16.1).
36
Unless otherwise noted, we let b1 = L1 = 0.
35
16
PR
Comparing the outcomes in the two cases, note that GOF
whenever bOF
1 > G1
2 (βκ −
PR
37
OF
PR
OF
PR
δ) > b2 (β − δ). Consequently, G1 > G1 implies b2 > b2 and thus, countries that
borrow from official sources tend to be more heavily indebted than countries borrowing
from private sources. To understand the country’s choice of debt instrument consider first
the case of δ = 0. We found above that the debt level corresponding to the maximum of
the debt-Laffer curve is higher in the corner with official debt. But this does not imply
that a myopic government that aims at maximizing the deficit necessarily chooses official
OF
over private debt since the former may be lower priced. In fact, comparing GPR
1 with G1
for δ = 0 reveals that the borrower will opt for official debt if and only if 1 − κ < L0 that
is, if the positive effect of stronger credibility on prices outweighs the mark-down. In the
following, we posit that this condition is met so that a myopic government would favor
issuing debt to official creditors.
In the range 0 ≤ δ ≤ βκ where both bPR
and bOF
are positive the criterion for the
2
2
choice of debt instrument is
(βκ − δ)2
1
(β − δ)2
1
OF
PR
.
−
G1 − G1 =
2f2 2βκ − δ(1 − L0 ) 1 − L0
2β − δ
For κ = 1, this expression is positive and official debt is preferred since it generates benefits
PR
of credibility at no cost. For κ < 1, GOF
is strictly positive for δ = 0, negative for
1 − G1
δ = βκ and convex in δ in-between implying that there exists a unique threshold value δ ?
such that for δ ≤ δ ? (high borrowing needs) official funding is preferred while for δ > δ ?
(low borrowing needs) private funding is preferred. The model thus predicts that episodes
of high borrowing needs (as captured by a low δ/β ratio) are associated with borrowing
from official rather than private sources.
PR
Figure 1 presents a numerical example. It plots the difference GOF
1 − G1 against δ for
β = 0.9 and f2 = 0.1. The solid curve corresponds to intermediate values of enforcement
PR
power (L0 = 0.25) and price discount (κ = 0.9). Holding δ fixed, the difference GOF
1 − G1
0
0
increases if L is raised (dashed curve for L = 0.4) and decreases if κ is lowered (dotted
curve for κ = 0.8). Stronger enforcement power therefore raises δ ? and renders official
funding more likely while higher price discounts lower δ ? and increase the relative advantage of private funding. Under the maintained assumptions, these intuitive comparative
statics results hold for arbitrary parameter combinations.
The default risk F2 (b2 − L(be2 )) along the equilibrium path is given by f2 bPR
= (β −
2
OF
δ)/(2β − δ) if the sovereign opts to borrow from private lenders and f2 b2 (1 − L0 ) =
(βκ − δ)/(2βκ − δ(1 − L0 )) if it borrows from official sources. As long as κ ≤ 1 the default
risk under private lending always exceeds the one under official lending. In equilibrium
(where the government chooses between the two sources of funding) default risk therefore
is weakly lower than if the sovereign only had the option to borrow from private lenders.
More favorable funding conditions offered by official lenders (higher values of κ) increase
the amount of borrowing from official sources and reduce the difference between the two
default rates.
Turning to prices, a given amount of debt, b2 , carries the price β(1 − f2 · b2 ) when
issued to private lenders and κβ(1 − f2 · b2 (1 − L0 )) when issued to official lenders. A given
37
PR
PR
This follows from the fact that GOF
= bOF
1
2 (βκ − δ)/2 and G1 = b2 (β − δ)/2.
17
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
Δ
0.2
0.4
PR
as function of δ. Higher L0 shifts the curve up (dashed line), lower
Figure 1: GOF
1 − G1
κ shifts the curve down (dotted line).
amount of debt therefore is cheaper when financed from official sources if and only if
L0 ≥
1 − f 2 b2 1 − κ
.
f 2 b2
κ
Strong enforcement power, large levels of debt and a small mark-down on official funds
(a large value for κ) all contribute to lowering the interest rate on official debt relative to
the one on private debt.
Endogenous Price Discount, No Long-Term Debt Overhang Consider next the
case where the price discount is determined endogenously as the outcome of bargaining
between the sovereign and the enforcer. In the simplest case, all bargaining power lies
with the sovereign and default generates a cost C(be2 ) to the enforcer (in addition to the
capital loss). The binding participation constraint of the enforcer (3) then reads
be2 p1 (s1 , π1 ) = be2 β(1 − F2 ) − βF2 C(be2 ),
where, as before, we let F2 ≡ f2 · (b2 − L0 be2 ). If the cost is linear, C 0 (be2 ) = C 0 ≥ 0, then
this participation constraint simplifies to
p1 (s1 , π1 ) = q1 (s1 , π1 ) − βF2 C 0 = β(1 − F2 )(1 + C 0 ) − βC 0 .
(15)
The properties of the equilibrium in this example are similar to those obtained previously. The equilibrium is in a corner. If sovereign debt is exclusively funded from private
sources, the level of debt and the value of the government’s program remain unchanged
relative to the previous example. But if all debt is funded from official sources then the
equilibrium (for δ ≤ β) is characterized by
bOF
2 =
β−δ
1
1
(β − δ)2
1
1
OF
=
,
G
.
1
f2 2β(1 + C 0 ) − δ(1 − L0 ) 1 − L0
2f2 2β(1 + C 0 ) − δ(1 − L0 ) 1 − L0
18
The maximum of the debt-Laffer curve now is at the debt level (2f2 (1 + C 0 )(1 − L0 ))−1 ,
the level chosen by a myopic government, and yields a default probability of 1/2(1 + C 0 ).
Consequently, as long as (1 + C 0 )(1 − L0 ) ≤ 1 and δ = 0, more debt is issued when the
source is official rather than private.
As far as the choice of the debt instrument in the range 0 ≤ δ ≤ β is concerned, the
desirability of official relative to private funds is determined by
1
1
1
β − δ OF
1
PR
2
OF
(β − δ)
=
(b2 − bPR
G1 − G1 =
−
2 )
0
0
0
2f2
2β(1 + C ) − δ(1 − L ) 1 − L
2β − δ
2
PR
and official funding is preferred if and only if bOF
2 ≥ b2 .
PR
OF
The difference G1 − G1 is positive at δ = 0 if (1 + C 0 )(1 − L0 ) ≤ 1, attains a zero
in the interval [0, β) if L0 (2C 0 + L0 ) < 2C 0 , and always attains a zero at δ = β. Hence, if
the first two conditions are satisfied, there exists a unique threshold value δ ? such that
for δ ≤ δ ? (high borrowing needs) official funding is preferred while the opposite holds for
δ > δ ? . The threshold value increases with L0 , as in the previous example, and falls with
C 0 . This is intuitive since a higher C 0 increases the expected costs (beyond capital losses)
that the enforcer bears in case of default; in order to compensate for these expected costs,
the enforcer requires a premium relative to the rate charged by private debt buyers. An
increase of C 0 therefore has the same qualitative effect on δ ? as a decrease of κ in the
previous example.
As far as the price of funds is concerned, a fixed quantity of debt b2 carries a higher
interest rate when raised from private sources. The price for such debt equals β(1 − f2 b2 )
while the price for the same quantity of debt issued to official creditors equals β(1 −
f2 b2 (1 − L0 ))(1 + C 0 ) − βC 0 (from (15)) which is larger than β(1 − f2 b2 ) under the first
condition described above.
These findings are robust to changing the specification of the cost function C(·). Suppose, for example, that costs are not proportional but contain a fixed component so that
C(be2 ) = c > 0 if be2 > 0 and C(be2 ) = 0 if be2 = 0. The enforcer’s participation constraint
(3) satisfied at equality then reads
p1 (s1 , π1 ) = q1 (s1 , π1 ) − βf2 · (b2 − L0 be2 )c/be2
PR
> 0 at
and equilibrium is again at a corner. Under conditions guaranteeing GOF
1 − G1
?
OF
PR
δ = 0, an increase in c reduces the threshold value δ at which G1 = G1 . That is, a
higher fixed cost c has the same qualitative effect on δ ? as higher variable costs C 0 or a
lower κ in the previous examples.38
Exogenous Price Discount, Long-Term Debt Overhang Finally, consider the consequences of long-term debt overhang, b02 ξ1 > 0, for the choice of debt instrument as well
38
We have (for δ ≤ β(1 − cf2 (1 − L0 )))
bOF
=
2
1 β(1 − cf2 (1 − L0 )) − δ 1
1 (β(1 − cf2 (1 − L0 )) − δ)2 1
, GOF
=
.
1
0
0
f2
2β − δ(1 − L )
1−L
2f2
2β − δ(1 − L0 )
1 − L0
19
as the decision to default in the first period. In parallel to the first example, we assume
an exogenous and constant mark-down, p1 (s1 , π1 ) = κq1 (s1 , π1 ). The marginal effects
M(b2 , be2 ) and Me (b2 , be2 ) then are unchanged relative to (13) and (14) except that the
probability of default, F2 , is given by f2 · (b̃2 − L0 be2 ) rather than f2 · (b2 − L0 be2 ). We also
assume that f2 b02 ξ1 < 1 so that the probability of default is smaller than one and new
debt issuance depresses the price of debt.
If sovereign debt is exclusively funded from private sources then the equilibrium level
of debt (for δ ≤ β) is given by
bPR
2 =
1 β−δ
(1 − f2 b02 ξ1 ).
f2 2β − δ
Less debt—by a factor of (1−f2 b02 ξ1 )—is issued relative to the case without long-term debt
overhang. This is due to the fact that outstanding long-term debt renders default more
likely, raising the (absolute value of the) price elasticity and reducing the attractiveness
of new debt issuance.
When all debt is financed by official sources then the equilibrium debt level (for sufficiently low values for δ) is given by
bOF
2 =
1
βκ − δ
1
δL0 f2 b02 ξ1
1
1
(1
−
f
b
ξ
)
−
.
2
02
1
0
0
0
f2 2βκ − δ(1 − L ) 1 − L
f2 2βκ − δ(1 − L ) 1 − L0
As in the case with private refinancing, outstanding long-term liabilities reduce the incentive to issue new debt because they push the borrowing country closer to the top of
the debt-Laffer curve. This effect is reflected in the wedge (1 − f2 b02 ξ1 ). But long-term
debt overhang carries an additional cost when accompanied by the issuance of official debt
because it increases the marginal expected cost of enforcer funds due to reduced future
flexibility, as discussed in the context of the general model. This lowers the optimal new
debt issuance, as reflected in the right-most term in the expression for bOF
2 .
If δ = 0 the costs due to reduced future flexibility are not internalized by the government in the first period. Debt overhang therefore affects the trade-off between official and
private funds only through its effect on the price elasticity. In particular, the difference
between the two values is given by
PR
GOF
1 − G1 |δ=0 = −
β(1 − κ − L0 )
(1 − f2 b02 ξ1 )
4f2 (1 − L0 )
such that official funds are preferred over private funds whenever 1 − κ < L0 (exactly
as without debt overhang). While sufficiently strong credibility gains and a low markdown lead the government to prefer official over private funds this relative attractiveness
decreases with the size of the debt overhang.
When δ > 0 the government internalizes costs due to reduced future flexibility and
debt overhang reduces the relative attractiveness of official refinancing for this second
PR
reason as well. Figure 2 plots the difference GOF
as a function of δ for different
1 − G1
values of b02 ξ1 . The parameter values are as in the example without long-term debt
overhang. We also assume that b1 = L1 = 0. The solid line corresponds to b02 ξ1 = 0,
20
the dashed line to b02 ξ1 = 1 and the dotted line to b02 ξ1 = 3. The figure shows that
official debt becomes less desirable (the threshold value of δ ? for the choice of official
debt becomes smaller) as outstanding debt increases. Intuitively, higher long-term debt
overhang places the economy closer to the top of the debt-Laffer curve where the benefits
from higher credibility are outweighed by the costs associated with reduced flexibility in
the future and the mark-down.
0.5
0.4
0.3
0.2
0.1
0.0
0.2
0.4
0.6
0.8
1.0
Δ
0.1
0.2
PR
as function of δ. Higher b02 ξ1 reduces δ ? .
Figure 2: GOF
1 − G1
Turning to the default decision in the first period, note that this decision is independent
of the stock of long-term debt overhang if the government can only default on currently
maturing debt (ξ1 = 1). In this case, the threshold value L̂1 at which it becomes optimal
to default equals L̂1 = b1 , as without outstanding debt.
If default applies to both maturing and outstanding debt (ξ1 = r1 ) then L̂1 exceeds b1
whenever b02 > 0 (see (9)) and the sovereign’s incentive to default increases with the stock
of outstanding long-term debt.39 As discussed above, this incentive can be particularly
strong if refinancing is provided from official sources because in this case default moves
the country away from the top of the debt-Laffer curve and also reduces the marginal
expected cost of enforcer funds due to reduced flexibility in the future.
Figure 3 illustrates this (ignore the solid curve for now). It displays the threshold
values L̂1 as a function of δ and conditional on refinancing from either of the two sources
of fresh funds. The default threshold L̂PR
applies when new debt must be financed by
1
private investors; and the threshold L̂OF
applies
when it must be issued to official lenders.
1
Default occurs for realizations of L1 below the relevant loci. For b02 = 0, the default
thresholds are independent of δ and the two loci coincide and are flat at level b1 . For
b02 > 0, as in the example illustrated in the figure (where b1 = 0, b02 = 2), the loci have
a non-zero slope because default reduces b02 r1 to zero, and the effect of this change on
the value of the government’s program depends on δ.40 More to the point, the figure
39
In two different environments, one with ξ1 = r1 and the other with ξ1 = 1, the cost of defaulting
might differ. We disregard such differences as they are irrelevant for our analysis because we do not
compare outcomes across environments.
40
A reduction of b02 r1 affects the price of new debt, the equilibrium quantity of debt issued as well as
21
shows that for low values of δ, intermediate realizations of L1 (for instance, L1 = 0.9 for
δ = 0) induce the sovereign to default if refinancing is provided from official but not if it
is provided from private sources.
L1
2.6
2.4
2.2
2.0
1.8
1.6
1.4
∆
0.2
0.4
0.6
0.8
1.0
(dashed), L̂1 (solid) as functions of δ.
(dotted), L̂OF
Figure 3: L̂PR
1
1
The solid line in figure 3 represents the equilibrium default threshold L̂1 as a function
of δ. It coincides with the default threshold conditional on official refinancing, L̂OF
1 ,
whenever the government chooses to borrow from official sources independently of the
realization of L1 . This is the case when δ is low such that funding needs are strong. With
a low realization of L1 , the country then defaults, long-term debt overhang is wiped out
and the benefits of credibility make it optimal to seek official funds. For high realizations
of L1 , in contrast, the country does not default and long-term debt overhang persists but
the country still prefers official funds because of the low importance (due to the low value
of δ) it attaches to future flexibility.
When δ is high the government’s choice of refinancing source also is independent of the
realization of L1 but in this case borrowing from private sources is always preferred. The
equilibrium default threshold therefore coincides with the default threshold conditional
on private refinancing. For intermediate values of δ (in an interval around 0.3), though,
default and refinancing decisions interact and the equilibrium default threshold differs
from both L̂OF
and L̂PR
1
1 . Default occurs less often than when the government can only
tap official funds but more often than in the case where the government can only borrow
from private sources. Low realizations of L1 and the ensuing default lead the government
to seek official funds with their credibility benefits while conditional on high realizations of
L1 and the ensuing non-default, the presence of honored long-term debt overhang renders
official refinancing unattractive. As a consequence, the choices of repayment rate and
refinancing source are fully correlated in this intermediate range of δ values.
Figure 4 summarizes how the default decision and the debt ownership structure vary
the amount of long- and short-term debt to be serviced in the future. The price, the quantity, and the
weight attached to the future all depend on δ.
22
with δ and L1 (ignore the dashed curves for now). The government defaults when L1
is small and it seeks refinancing from official sources for low values of δ. The solid
demarcation lines in the figure identify four resulting policy regions in (δ, L1 )-space. The
dashed demarcation lines identify the same policy regions under the assumption of a higher
debt overhang (b02 = 3 instead of b02 = 2). The increase in the stock of outstanding longterm debt has two effects on the regions. First, it increases the horizontal demarcation
line separating the regions with r1 = 0 from those with r1 = 1. Intuitively, a default
improves both the price elasticity of fresh funds and the continuation value more strongly
if the debt overhang is higher, independently of the source of refinancing. Second, higher
debt overhang shifts to the left the upper part of the vertical demarcation line separating
the top-left and top-right policy regions (r1 = 1 and be2 > 0 versus r1 = 1 and be2 = 0).
Intuitively, when default is too costly higher debt overhang reduces the attractiveness
of official relative to private refinancing, both because of the higher price elasticity and
the higher marginal expected cost of enforcer funds due to reduced future flexibility (see
figure 2 and the discussion surrounding it).
Figure 5 illustrates the equilibrium debt issuance in the first period, b2 . New debt
issuance is higher after a default (low realizations of L1 ) because the latter eliminates
long-term debt overhang and thereby improves access to new funding. Default therefore
increases consumption both directly and indirectly. With a higher stock of outstanding
long-term debt (case not depicted in the figure) new debt issuance falls for high values
of L1 because the government does not default in those states and higher debt overhang
renders new funding less attractive. For low values of L1 the quantity and price of new
funds does not change relative to the situation with lower debt overhang because the
overhang is wiped out anyway.
L1
3.0
2.5
2.0
r1 = 1, be2 > 0
r1 = 1, be2 = 0
r1 = 0, be2 > 0
r1 = 0, be2 = 0
1.5
1.0
0.5
∆
0.0
0.2
0.4
0.6
0.8
1.0
Figure 4: Default and official lending regions for b02 = 2 (solid) and b02 = 3 (dashed).
In sum, figures 3–5 illustrate how default and refinancing decisions interact and more
specifically, how the availability of official funding may increase default risk on outstanding debt. In fact, figure 4 lends itself to a compact characterization of country experiences
23
Figure 5: Total borrowing as function of δ and L1 .
during the recent Eurozone sovereign debt crisis. Consider a sudden increase in refinancing needs, indicated by a fall of δ below the lower part of the vertical demarcation line.
The model then predicts the following: First, countries that choose to default receive refinancing from official sources. The experience of Greece supports this prediction. Second,
countries that choose not to default may seek refinancing from official or private sources,
with heavily indebted countries (high debt overhang) less likely to opt for official funds.
The experiences of Ireland, Italy and Portugal are consistent with this prediction while
the experience of Spain may be considered consistent or not.41
Finally, the model has another, somewhat provocative implication. As emphasized
several times, the availability of official funding may increase default risk on outstanding
debt when refinancing needs are high. Interestingly, it may not only be the borrowing
country that favors default in these circumstances. Official creditors may favor it as well
since they profit from debt they buy as long as κ < 1 and, as a consequence, from any
increase in the demand for official funds.
Consequences of Relaxing the Linearity Assumptions Relaxing the linearity assumptions may give rise to an interior equilibrium debt composition. For example, maintaining the assumption of linear utility and a uniform distribution function but letting
L(·) be strictly concave rather than linear produces a strictly concave objective function
G1 (s1 ) if κ < 1. Depending on parameter values the short-selling constraints may still
41
At the time when official funds were provided, the debt quotas of Ireland and Portugal (which
received refinancing from official sources) were lower than the quota of Italy which did not receive such
refinancing. Spain had an even lower debt quota. While it has not actually received official funds the
Eurogroup committed to provide up to 100 billion Euro. IMF (2012, p. 17) contains deficit and debt
quota data.
24
bind in this case and induce a corner solution. But if they do not bind, the model predicts an interior equilibrium debt composition, in addition to the interior total quantity
of borrowing.
5
Concluding Remarks
During the recent sovereign debt crisis in the Eurozone, patterns pertaining to default
decisions and the composition of debt have varied significantly across debt distressed
countries. While Greece defaulted and sought refinancing from official sources, Ireland,
Portugal and Spain did not default but still drew official funds (or accepted a Eurogroup
commitment for conditional support), and Italy neither defaulted nor relied on official
loans. We have proposed a simple theory that helps shed light on these patterns.42
Our model adds two features to the standard sovereign debt framework: Differential
enforcement powers across groups of creditors, and the pari passu provision. These features combine to produce default and debt composition decisions that seem consistent
with the empirical evidence, for example default accompanied by “inexpensive” official
lending; substitution of official for private debt in times of debt distress, with a tendency
towards larger debt quantities when funds are drawn from official sources; crowding out
of private debt even under pari passu; and resistance to draw official funds at favorable
rates by troubled but non-defaulting countries with large long-term debt obligations. Another prediction of the model, namely that official would be creditors may encourage a
sovereign to default on outstanding private debt is unexpected, yet equally plausible under
the assumption that these creditors are in a position to extract rents.
An advantage of our model lies in its simplicity. In particular, the setup generates
cross-country variation of government choices without incorporating political economy
considerations. But this omission also represents an opportunity to expand the model’s
explanatory power by including such relevant elements. Future research could attempt
this fusion. It could also address normative questions, for example with respect to the
welfare implications of pari passu provisions or, more generally, the optimal seniority
structure of official and private loans.
Finally, note that our analysis is quite general and applies equally well to credit relationships that do not involve sovereign debt as long as classes of creditors differ in terms
of the punishment they can inflict on delinquent debtors.
42
For a recent view related to the one proposed in this paper, see “The eurozone’s journey to defaults,”
Financial Times, March 11, 2011.
25
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